Problem: Simplify and expand the following expression: $ \dfrac{4}{5p - 25}- \dfrac{5}{4p + 36}- \dfrac{3p}{p^2 + 4p - 45} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{4}{5p - 25} = \dfrac{4}{5(p - 5)}$ We can factor a $4$ out of denominator in the second term: $ \dfrac{5}{4p + 36} = \dfrac{5}{4(p + 9)}$ We can factor the quadratic in the third term: $ \dfrac{3p}{p^2 + 4p - 45} = \dfrac{3p}{(p - 5)(p + 9)}$ Now we have: $ \dfrac{4}{5(p - 5)}- \dfrac{5}{4(p + 9)}- \dfrac{3p}{(p - 5)(p + 9)} $ The least common multiple of the denominators is: $ 20(p - 5)(p + 9)$ In order to get the first term over $20(p - 5)(p + 9)$ , multiply by $\dfrac{4(p + 9)}{4(p + 9)}$ $ \dfrac{4}{5(p - 5)} \times \dfrac{4(p + 9)}{4(p + 9)} = \dfrac{16(p + 9)}{20(p - 5)(p + 9)} $ In order to get the second term over $20(p - 5)(p + 9)$ , multiply by $\dfrac{5(p - 5)}{5(p - 5)}$ $ \dfrac{5}{4(p + 9)} \times \dfrac{5(p - 5)}{5(p - 5)} = \dfrac{25(p - 5)}{20(p - 5)(p + 9)} $ In order to get the third term over $20(p - 5)(p + 9)$ , multiply by $\dfrac{20}{20}$ $ \dfrac{3p}{(p - 5)(p + 9)} \times \dfrac{20}{20} = \dfrac{60p}{20(p - 5)(p + 9)} $ Now we have: $ \dfrac{16(p + 9)}{20(p - 5)(p + 9)} - \dfrac{25(p - 5)}{20(p - 5)(p + 9)} - \dfrac{60p}{20(p - 5)(p + 9)} $ $ = \dfrac{ 16(p + 9) - 25(p - 5) - 60p} {20(p - 5)(p + 9)} $ Expand: $ = \dfrac{16p + 144 - 25p + 125 - 60p}{20p^2 + 80p - 900} $ $ = \dfrac{-69p + 269}{20p^2 + 80p - 900}$